Abstract

In the scalar case, the spectral factorization mapping \(f\rightarrow f^+\) puts a nonnegative integrable function f having an integrable logarithm in correspondence with an outer analytic function \(f^+\) such that \(f = |f^+|^2\) is almost everywhere. The main question addressed here is to what extent \(\Vert f^+ - g^+\Vert _{H_2}\) is controlled by \(\Vert f-g\Vert _{L_1}\) and \(\Vert \log f - \log g\Vert _{L_1}\).

Keywords

Spectral factorization Paley–Wiener condition Convergence rate

Lasha Ephremidze was partially supported by the Shota Rustaveli National Science Foundation Grant (Contract Numbers: 31/47) and Ilya Spitkovsky was supported in part by Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi, and by Plumeri Award for Faculty Excellence from the College of William and Mary.

where \(f^+\) is a function analytic inside the unit circle, \(f^+\in \mathcal {A}(\mathbb {T}_+)\), and \(f^-(z)=\overline{f^+(1/\overline{z})}\), which is analytic outside the unit circle including the infinity, \(f^-\in \mathcal {A}(\mathbb {T}_-)\). More specifically, \(f^+\) belongs to the Hardy space \(H_2(\mathbb {D})\); therefore, its boundary values \(f^+(t)=f^+(e^{i\theta })=\lim _{r\rightarrow 1}f^+(re^{i\theta })\) exist a.e. and the Eq. (2) holds for these boundary values. Note also that \(f^+=\overline{f^-}\) a.e. on \(\mathbb {T}\) and therefore (2) is equivalent to

Condition (1) is necessary for factorization (2) to exist. It also plays an important role in the linear prediction theory of stationary stochastic processes, one of the historically first applications of spectral factorization (see [16, 21]). Namely, let \(\ldots , X_{-1}, X_0, X_1,\ldots \) be a stationary stochastic process with the spectral measure \(d\mu =f\,dt+d\mu _s\). In a different but equivalent language, \(\{X_n\}_{n\in \mathbb {Z}}\) is a sequence in a Hilbert space and \(\langle X_n,X_k\rangle =\frac{1}{2\pi }\int _\mathbb {T} e^{i(n-k)\theta }\,d\mu (\theta )\). The process is deterministic if \(X_{n+1}\) can be represented as the limit of linear combinations of vectors from \(\{\ldots , X_{n-1}, X_n\}\), i.e, \(X_{n+1}\in \overline{\text{ Span }}\{\ldots , X_{n-1}, X_n\}\). As it happens (see e.g., [16]), condition (1) is necessary and sufficient for the process to be non-deterministic.

If we require \(f^+\) to be an outer analytic function, then the factorization (2) is unique up to a constant factor c with absolute value 1, \(|c|=1\). The unique spectral factor which is positive at the origin can be a priori written as

In most applications, a spectral factor \(f^+\) in (2) is not explicitly required to be outer and instead is subject to certain extremal conditions called, in various works, minimal phase or maximal energy, optimal, etc. In mathematical terms, however, they amount to \(f^+\) being outer, so seeking the solution (3) is natural.

From the practical point of view, it is important to study the continuity properties of the spectral factorization map

$$\begin{aligned} f\mapsto f^+ \end{aligned}$$

(4)

defined by (3). Namely, we are interested in knowing how close \(g^+\) is to \(f^+\) when a spectral density g is close to f. The reason why we study this question is that usually an estimated spectral density function \(\hat{f}\) being dealt with is constructed empirically from observations and is only an approximation to the theoretically existing spectral density f. Therefore, we need to know how close \(\hat{f}^+\) remains to \(f^+\) under such approximation.

An answer to the above question depends on norms we use as a measurement of the accuracy in the spaces of functions and of their spectral factors. Since the boundary values of the function (3) can be expressed as

and the conjugation is not a bounded operator on \(L_\infty \) or \(C(\mathbb {T})\), it is not surprising that the map (4) is not continuous in these spaces [1]. Furthermore, it is shown in [5] that every continuous function on \(\mathbb {T}\) is a discontinuity point of the spectral factorization mapping in the uniform norm, whereas in [14] it was shown that on a large class of function spaces (the so-called decomposing Banach algebras) the spectral factorization mapping is continuous.

i.e., the spectral factor \(f^+\) is a polynomial of the same degree N. This result is known as the Fejér-Riesz lemma (see, e.g., [8]). The spectral factor can also be expressed in terms of zeros of polynomial (5), and therefore the map (4) is continuous on \(\mathcal {P}_N\), the set of all functions of the form (5). Papers [6, 7] are devoted to estimating the constant \(C_N\) in the inequality

and it is shown there that \(C_N\sim \log N\) asymptotically, under the condition that the values of functions \(\phi \) and \(\psi \) are bounded away from 0.

Moving to Lebesgue spaces, the map (4) is not continuous in the \(L_1\) norm in general, since a small change of values of function f, if these values are close to 0, may cause a significant change of \(\log f\). However,

A proof of an analog of (6) for more general matrix case can be found in [3] or [10]. In the present paper, we discuss quantitative estimates of the rate in the above convergence. Firstly, we look for estimates of \(\Vert g^+-f^+\Vert _{H_2}\) in terms of \(\Vert g-f\Vert _{L_1}\) and \(\Vert \log g-\log f\Vert _{L_1}\). It turns out that, in general, there is no such estimate. Namely, there is no function \(\Pi : [0, +\infty )^2 \rightarrow [0, +\infty )\) such that \(\lim _{s, t \rightarrow 0} \Pi (s, t) = 0\) for which the estimate

Nevertheless, one can still obtain an estimate for \(\Vert g^+ - f^+\Vert _{H_2}\) if one takes into account that for each \(f \in L_1(\mathbb {T})\) there exists an Orlicz space \(L_\Psi (\mathbb {T})\) such that \(f \in L_\Psi (\mathbb {T})\) (see, e.g., [17, Sect. 8]). One can show that there exists a function \(\Pi _\Psi : [0, +\infty ) \rightarrow [0, +\infty )\) such that \(\lim _{t \rightarrow 0} \Pi _\Psi (t) = 0\) and

where \(u:[0,\infty )\longrightarrow [0,\infty )\) is a right continuous, nondecreasing function with \(u(0)=0\) and \(u(\infty ):=\lim _{t\rightarrow \infty }u(t)=\infty \), and v is defined by the equality \(v(x)=\sup _{u(t)\le x}t\). Let \((\Omega , \Sigma , \mu )\) be a measure space, and let \(L_\Phi (\Omega )\), \(L_\Psi (\Omega )\) be the corresponding Orlicz spaces, i.e., \(L_\Phi (\Omega )\) is the set of measurable functions on \(\Omega \) for which either of the following norms

If \(\Phi '\) is continuous, the above definition of \(\Lambda _\Phi \) can be rewritten in terms of inverse functions, because \(\Phi '\) is nondecreasing. For an arbitrary N-function \(\Phi \), one has

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